Re: Linear transformation help

There is NO question in your attachment, just what looks like the solution to a problem. I presume that they know that T(1)= -t and T(t)= 1 because of the way T is defined in the problem itself. If you were to post that, we might be able to point out exactly where it gives the definition of T.

Re: Linear transformation help

To find a matrix representation of a linear transformation in a given basis, apply the linear transformation to each basis vector in turn, the write the results as a linear combination of the basis vectors. The coefficients of the linear combinations give the columns of the matrix.

Here, the first basis is "{1, t}". As we saw before, T(1)= -t= 0(1)+ (-1)t and T(t)= 1(1)+ 0(t). The matrix representation in this basis is . In this basis we would represent the general "a+ bt" as the vector and you can see that which represents b- at.

Do the same for the basis {1+ t, 1- t}. T(1+ t)= 1+ t, since a= b= 1 and T(1- t)= -1+ t since a= -1 and b= 1. Of course, 1+ t= 1(1+ t)+ 0(1- t) and -1+ t= 0(1+t)- 1(1- t) so the matrix representation in this basis is .